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Abstract
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In this talk, we introduce and analyze some two-grid methods for nonlinear elliptic eigenvalue problems of the form $-\mbox{div}(A\nabla u)+Vu+f(u^2)u=\lambda u, \|u\|_{L^2}=1$. We provide a priori error estimates for the ground state energy, the eigenvalue $\lambda$, and the eigenfunction $u$, in various Sobolev norms. In particular we focus on the $\mathbb{P}_1$ and $\mathbb{P}_2$ finite element discretizations, and on the Fourier spectral approximation (for periodic problems), taking numerical integration error into account. Finally we provide some numerical examples to illustrate our analysis.
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